The eye has a shutter

Everything on this site treats a light as a number. A spectral power distribution is a function of wavelength and of nothing else; an illuminant is a constant; a chromaticity is a point. That is exactly right for daylight and for a hot filament, whose thermal mass makes them steady on any timescale a person has, and it is wrong for almost every light manufactured since.

The correction is not to the spectrum. A spectroradiometer averages, so the time-averaged spectrum every other page here computes with is unaffected, and so is every essay that uses one. What changes is whether anybody sees a steady light — and the answer turns on two functions that give out at frequencies four times apart.

The claim #

The eye is a filter in time as well as in space, the two channels have two cutoffs, and the frequency at which either gives out is a function of the light level rather than a property of the observer.

Four measurements, and the second and fourth are the ones with consequences:

• Luminance flicker fuses at about sixty hertz at an office light level. Familiar, and the number is a landmark rather than a derivation.
• Isoluminant flicker fuses at about fifteen — four times sooner. At thirty hertz the luminance channel is still at 0.373 of its peak and the chromatic one is at 3.7 × 10⁻⁴. So there is a whole band of frequencies at which a colour changing is invisible and the same modulation carried as brightness is not.
• Fusion is linear in the logarithm of the light, at 12.5 hertz per decade. A drive that fuses at ten hertz in starlight fuses at eighty-five in daylight; the same lamp changes verdict when somebody opens a curtain.
• And above fusion the eye integrates exactly. A flickering light matches a steady one of the same time average, which is Talbot’s law — and for a source whose three channels are pulsed with three different duty cycles, that average is a colour the source emits at no instant. The nearest colour it does emit is ΔE00 31.2 away.

The colour nobody emits #

The last of those is worth taking first, because it is the one that belongs to this site rather than to the flicker literature.

A three-primary source pulsed with duty cycles of 1, 0.5 and 0.25 emits, at every instant, one of three colours: all three channels for a quarter of the cycle, two for a quarter, one for a half. Above fusion the visual system integrates over the cycle, so what is seen is the time-weighted average of those three points.

One property of that average is worth stating before its size, because it bounds what the effect can do. The fused colour is a convex combination of the three instantaneous ones, weighted by their shares of the cycle — so it lies inside the triangle they span and can never be outside the source’s own gamut. A pulsed source does not reach colours it could not reach steadily; it reaches one of them by a route that passes through none of them.

The average of a set of points is not a member of the set. The colour delivered is a colour the source never emits, and the gap is not small: the nearest instantaneous colour is 31.2 ΔE00 away from the fused one, which is roughly the distance between a saturated orange and a mid grey.

This is not a curiosity of a contrived source. It is how every colour-sequential projector works — one channel at a time through a spinning wheel, whose primaries are three numbers like any others — and it is why a fast eye movement across such a projector’s image separates the picture into coloured fringes. The fringes are the instants, arriving on different parts of the retina, and the steady picture is the average that appears when they arrive on the same part.

Talbot’s law is exact in this model, and asserted to be. A pulsed source integrated over one cycle returns the level it was built around, to twelve decimal places — an exactness that is about the radiance rather than about the appearance, which is the distinction two sections below. That exactness is what makes the colorimetry usable at all above fusion: the average spectrum is a real spectrum, and every integral this site computes with it is right.

Why the colour channel gives out first #

An isoluminant modulation is a modulation of the chromatic opponent channels with the luminance channel held still. Those channels are slower, and the reason is the same one that makes them give out sooner in space: they are built by differencing cone signals, and differencing throws away the fast common part.

The ratio of four between the two cutoffs is the temporal counterpart of the ratio of four to six between the spatial ones. Both come out of the same architecture, both are measured on the same kind of apparatus, and neither is in any colorimetric standard.

The practical form of it: a display refreshing at forty hertz will show visible luminance flicker and no visible colour flicker at all, however violently its channels are being switched, provided their sum is steady. That is exactly what colour-sequential displays and pulse-width-modulated backlights exploit, and it is why the specification a panel is sold on names a refresh rate and never names a chromatic one.

Doubling the drive frequency moves the whole spectrum up past the band the luminance channel can follow, and nothing else about the waveform changes.

What Talbot’s law is asserting, in lightness #

“A flickering light matches a steady one of the same time average” sounds like a statement of the obvious, and it is not. The average it names is an average of radiance, and the alternative — an average of what each instant looks like — gives an answer thirty lightness units away.

A lamp at fifty per cent duty and full modulation is emitting full power for half the cycle and nothing for the other half. Talbot’s law says it is seen as a steady light at half power, whose lightness is 76.1. Averaging the two instants’ lightnesses gives 50, because the dark instant is at zero.

So Talbot’s law is a thirty-unit claim about the order of two operations, and it is the same claim the halftone essay makes about space: the pooling happens before the compression, so the eye computes the lightness of an average rather than the average of some lightnesses. Both cases are concave functions of a two-level signal, both have a Jensen gap of about the same size, and the gap peaks near a quarter coverage in one and a quarter duty cycle in the other for the same reason — the cube root’s curvature is steepest down where the dark level sits.

The two cases come out opposite ways round, and that is the interesting part. A halftone is a spatial two-level signal that the eye can partly resolve, so the compression acts on a pattern that is still black and white and the reader gets something nearer the average of the lightnesses. A flickering light above fusion is a temporal two-level signal the eye cannot resolve at all, so the integration is complete and the reader gets the lightness of the average.

Which means the two essays’ findings are one finding with a resolvability parameter in it. Where the structure is resolved, the nonlinearity acts first and the observer sees the pooled appearance; where it is not, the pooling acts first and the observer sees the appearance of the pooled stimulus. A halftone above thirty cycles per degree behaves like a flickering light above fusion, and both agree with the instrument; below their respective thresholds neither does, and each is out by about thirty lightness units in the same direction.

That also says what makes pulse-width dimming honest and halftone measurement fragile. A lamp is always above fusion by design — nobody ships a driver at ten hertz — so the temporal case is permanently in the regime where the instrument’s average is the right answer. A halftone is not always above the eye’s spatial cutoff, because the cutoff is an angle and the reader chooses the distance. One of the two structures gets to specify its own resolvability and the other does not.

Fusion is not a number #

The most repeated statement about flicker is that the eye fuses above about sixty hertz. It is a fair summary of one light level and it is not a property of a person.

Ferry–Porter: fusion frequency is linear in the logarithm of retinal illuminance. At the office level the model is anchored at, the criterion puts it at sixty hertz; a decade brighter it is 72.5, a decade dimmer 47.5.

The span is enormous compared with the tolerances anybody writes. A hundred-hertz drive — the flicker rate of every unfiltered lamp on a fifty-hertz supply — is comfortably fused in a living room and marginal in bright sunlight, and it is the same lamp.

Two numbers a lamp is sold by, and what each cannot see #

A lamp’s temporal behaviour is reported as percent flicker — the Michelson contrast of its waveform — and sometimes as a flicker index, the share of one cycle’s area sitting above the cycle’s own mean. The second exists because the first cannot tell a narrow spike from a square wave.

Two drives switching a diode fully on and fully off have the same maximum and the same minimum whatever their duty cycle, so both read 99.8 per cent flicker. Their indices are 0.50 and 0.89. A dimmer that works by shortening the pulse therefore holds its percent flicker constant at every setting and changes what is actually delivered, which is why a lamp can be reported as unchanged by dimming and be visibly worse at ten per cent than at full.

Neither number contains a frequency, and the frequency is what decides everything — the same shape of omission as a colour tolerance that names no arrangement. A hundred per cent modulation at fifty hertz is twenty-two times over threshold and a hundred per cent modulation at two hundred hertz is a hundredth of it — the same two figures of merit, the same lamp, and one of them is a defect.

What was computed, and how #

Five landmarks, and the functions are solved from them. Where the luminance function peaks, where each channel fuses, how deep the low-frequency dip is, the contrast threshold at the peak, and the Ferry–Porter slope. Each is stated with the range the literature reports, and the two transfer functions are the solution of a small nonlinear system rather than a curve drawn through the points — the same construction, and the same damped Newton, the spatial file uses. The solve reproduces all three luminance constraints to 10⁻⁹, which is the check that the solver actually solved.

A waveform is decomposed before it is judged. A square wave is not one frequency: a hundred-hertz drive at fifty per cent duty carries a third of its fundamental at three hundred hertz and a fifth at five hundred. Every harmonic is put through the sensitivity function and the largest response decides, which is the same construction the spatial file uses for a pattern.

And the fused colour is an integral over the cycle, sampled at 720 instants, of spectra built from three Gaussian primaries — not of three hardcoded colours. The swatches are computed the way every swatch on this site is: a spectrum, an observer, a space, and a gamut check.

A sixty-hertz drive with a narrow duty cycle is the worst case a mains-powered lamp can present, and it is worth drawing on its own.

Where the model stops #

There is no adaptation state. The temporal sensitivity function used here is measured at one light level and the Ferry–Porter law is applied on top of it as a shift of the cutoff. A full treatment would move the whole curve’s shape with the level, and the peak frequency does move in the measurements.

The chromatic threshold is not commensurable with the luminance one. Both channels are anchored at the same absolute threshold contrast, which is the same deliberate simplification the spatial model states: chromatic temporal thresholds are measured in cone-contrast units that do not convert. What this essay compares is frequencies, which the anchor does not touch.

There is no eye of finite size. Flicker is more visible in the periphery than at the fixation point — the reason a fluorescent tube flickers most when it is not being looked at — and this model has no position in it at all.

And there is no non-linearity. A real visual system rectifies, so a deep modulation produces a response at twice its frequency that a linear filter has no term for.

Four hundred hertz is where a well-designed driver runs, and reducing the modulation with it is the pair of decisions a designer actually makes.

The generalisation #

The sentence worth carrying: a colour is an average, and the thing being averaged over is not only wavelength.

Colorimetry integrates over wavelength and treats the result as the stimulus. Above fusion it also integrates over time, which is why the arithmetic still works — but the fact that it works is a fact about the observer rather than about the light, and it fails below fifteen hertz for colour and below sixty for brightness.

The surprising connection is with metamerism. Two spectra that integrate to the same three numbers are a metameric pair; a pulsed source and a steady one of the same average are a temporal metameric pair, matched by the same kind of integration and broken by the same kind of change — not a change of illuminant this time, but a change of light level, which moves the fusion frequency and can put a source back on the wrong side of it. A pair of lamps matched in a bright shop and mismatched in a dim room is the ordinary illuminant-metamerism story; a lamp that is steady in a bright shop and flickers in a dim room is the temporal one, and nothing in the vocabulary of colour science names it.

Who found it, and when #

Talbot’s law is 1834, and Plateau’s independent statement is 1835: above fusion, a flickering light matches a steady one of the same time average. It was a claim about a spinning disc with sectors cut out of it, and it is still exactly the statement that makes pulse-width dimming a colorimetrically honest technique.

Ferry and Porter established the logarithmic law in 1892 and 1902 respectively, on apparatus that would not have been out of place in a workshop.

The chromatic–luminance split is later and is de Lange’s, from the 1950s, with the isoluminant measurements refined through the following decades. The finding that the chromatic channels are low-pass where the luminance channel is band-pass is the temporal counterpart of the spatial result and was arrived at independently.

What has not changed is practice. A display is specified by a refresh rate, a lamp by a percent flicker, and neither number carries the light level that decides whether it matters.

Where the mains is sixty hertz the rectified supply arrives at a hundred and twenty, and the comparison of three waveforms is what shows why one number cannot describe any of them.

What the pictures cannot show #

A page cannot flicker. Every figure here is a drawing of a waveform and of what a model says about it. The reader’s own experience of a lamp is the evidence, and the honest form of every claim on this page is conditional: at this light level, this drive is over or under threshold.

And the swatches in the fused-colour figure are the most misleading thing on the page. They are drawn side by side, steady, on a display that is itself pulsing. What the figure claims is that a source alternating between the three left-hand colours is seen as the right-hand one — and it makes that claim by showing four steady patches, which is the opposite arrangement.

Where the ladder goes next #

The nearest unfinished piece is the join to motion, and the next essay in this field takes it: a temporal modulation on something moving becomes a spatial pattern, at f divided by the speed, and the spatial sensitivity function this site already has answers whether it is visible. That derivation puts the visible ceiling for a flickering lamp three orders of magnitude above fusion, without fitting anything.

The second is the join to adaptation. The model has no clock measured how long an appearance judgement takes to settle; this measures how fast a stimulus can change before the eye stops following it. They are the same axis at two ends — seconds and milliseconds — and nothing connects them.